Volume And Surface Area Of Solids
Class 8th Mathematics RS Aggarwal Solution
- Find the volume, lateral surface area and the total surface area of the cuboid…
- The dimensions of a rectangular water tank are 2 m 75 cm by 1 m 80 cm by 1 m 40…
- A solid rectangular piece of iron measures 1.05 m x 70 cm x 1.5 cm. Find the…
- The area of a courtyard is 3750 m^2 . Find the cost of covering it with gravel…
- How many persons can be accommodated in a hall of length 16 m, breadth 12.5 m…
- A cardboard box is 1.2 m long, 72 cm wide and 54 cm high. How many bars of soap…
- The size of a matchbox is 4 cm x 2.5 cm x 1.5 cm. What is the volume of a…
- How many planks of size 2 m x 25 cm x 8 cm can be prepared from a wooden block…
- How many bricks, each of size 25 cm x 13.5 cm x 6 cm, will be required to build…
- A wall 15 m long, 30 cm wide and 4 m high is made of bricks, each measuring 22…
- Find the capacity of a rectangular cistern in litres whose dimensions are 11.2…
- The volume of a block of gold is 0.5 m^3 . If it is hammered into a sheet to…
- The rainfall recorded on a certain day was 5 cm. Find the volume of water that…
- A river 2 m deep and 45 m wide is flowing at the rate of 3 km/h. Find the…
- A pit 5 m long and 3.5 m wide is dug to a certain depth. If the volume of…
- A rectangular water tank is 90 cm wide and 40 cm deep. If it can contain 576…
- A beam of wood is 5 m long and 36 cm thick. It is made of 1.35 m^3 of wood.…
- The volume of a room is 378 m^3 and the area of its floor is 84 m^2 . Find the…
- A swimming pool is 260 m long and 140 m wide. If 54600 cubic metres of water…
- Find the volume of wood used to make a closed box of outer dimensions 60 cm x…
- Find the volume of iron required to make an open box whose external dimensions…
- A box with a lid is made of wood which is 3 cm thick. Its external length,…
- The external dimensions of a closed wooden box are 62 cm, 30 cm and 18 cm. If…
- A closed wooden box 80 cm long, 65 cm wide and 45 cm high, is made of…
- Find the volume, lateral surface area and the total surface area of a cube…
- The surface area of a cube is 1176 cm^2 . Find its volume.
- The volume of a cube is 729 cm^3 . Find its surface area.
- The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm. It is melted and…
- If the length of each edge of a cube is doubled, how many times does its…
- A solid cubical block of fine wood costs 256 at 500 per m^3 . Find its volume…
- Find the volume, curved surface area and total surface area of each of the…
- A milk tank is in the form of a cylinder whose radius is 1.5 m and height is…
- A wooden cylindrical pole is 7 m high and its base radius is 10 cm. Find its…
- Find the height of the cylinder whose volume is 1.54 m^3 and diameter of the…
- The volume of a circular iron rod of length 1 m is 3850 cm^3 . Find its…
- A closed cylindrical tank of diameter 14 m and height 5 m is made from a sheet…
- The circumference of the base of a cylinder is 88 cm and its height is 60 cm.…
- The lateral surface area of a cylinder of length 14 m is 220 m^2 . Find the…
- The volume of a cylinder of height 8 cm is 1232 cm^3 . Find its curved surface…
- The radius and height of a cylinder are in the ratio 7 : 2. If the volume of…
- The curved surface area of a cylinder is 4400 cm^2 and the circumference of…
- A particular brand of talcum powder is available in two packs, a plastic can…
- Find the cost of painting 15 cylindrical pillars of a building at Rs. 2.50 per…
- A rectangular vessel 22 cm by 16 cm by 14 cm is full of water. If the total…
- A piece of ductile metal is in the form of a cylinder of diameter 1 cm and…
- A solid cube of metal each of whose sides measures 2.2 cm is melted to form a…
- How many cubic metres of earth must be dug out to sink a well which is 20 m…
- A well of inner diameter 14 m is dug to a depth of 12 m. Earth taken out of it…
- A road roller takes 750 complete revolutions to move once over to level a…
- A cylinder is open at both ends and is made of 1.5-cm-thick metal. Its…
- The length of a metallic tube is 1 metre, its thickness is 1 cm and its inner…
- The maximum length of a pencil that can be kept in a rectangular box of…
- The total surface area of a cube is 150 cm^2 . Its volume isA. 216 cm^3 B. 125…
- If the capacity of a cylindrical tank is 1848 m^3 and the diameter of its base…
- The volume of a cube is 343 cm^3 . Its total surface area isA. 196 cm^2 B. 49…
- The cost of painting the whole surface area of a cube at the rate of is 10…
- How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to…
- How many cubes of 10 cm edge can be put in a cubical box of 1 m edge?A. 10 B.…
- The edges of a cuboid are in the ratio 1: 2: 3 and its surface area is 88 cm2.…
- Two cubes have their volumes in the ratio 1 : 27. The ratio of their surface…
- The surface area of a (10 cm x 4 cm x 3 cm) brick isA. 84 cm^2 B. 124 cm^2 C.…
- An iron beam is 9 m long, 40 cm wide and 20 cm high. If 1 cubic metre of iron…
- A rectangular water reservoir contains 42000 litres of water. If the length of…
- The dimensions of a room are (10 mx8mx 3.3 m). How many men can be…
- A rectangular water tank is 3 m long, 2 m wide and 5 m high. How many litres…
- The area of the cardboard needed to make a box of size 25 cm x 15 cm x 8 cm…
- The diagonal of a cube measures 4 root 3 cm. Its volume isA. 8 cm^3 B. 16 cm^3…
- The diagonal of a cube is 9 root 3 cm long. Its total surface area isA. 243…
- If each side of a cube is doubled then its volumeA. is doubled B. becomes 4…
- If each side of a cube is doubled, its surface areaA. is doubled B. becomes 4…
- Three cubes of iron whose edges are 6 cm, 8 cm and 10 cm respectively are…
- Five equal cubes, each of edge 5 cm, are placed adjacent to each other. The…
- A circular well with a diameter of 2 metres, is dug to a depth of 14 metres.…
- The ratio of the total surface area to the lateral surface area of 20 cm and…
- The number of coins, each of radius 0.75 cm and thickness 0.2 rightcm, to be…
- 66 cm^3 of silver is drawn into a wire 1 mm in diameter. The length of the…
- The height of a cylinder is 14 cm and its diameter is 10 cm. The volume of the…
- The height of a cylinder is 80 cm and the diameter of its base is 7 cm. The…
- The height of a cylinder is 14 cm and its curved surface area is 264 cm^2 .…
- The diameter of a cylinder is 14 cm and its curved surface area is 220 cm^2 .…
- The ratio of the radii of two cylinders is 2 : 3 and the ratio of their…
- Find the volume of a cube whose total surface area is 384 cm^2 .
- How many soap cakes each measuring 7 cm 7cmx 5 cm x 2.5 cm can be placed in a box of…
- The radius and height of the cylinder are in the ratio 5 : 7 and its volume is 550 cm^3…
- Find the number of coins, 1.5 cm in diameter and 0.2 cm thick, to be melted to form a…
- Find the surface area of a chalk box, whose length, breadth and height are 18 cm, 10 cm…
- The curved surface area of a cylindrical pillar is 264 m^2 and its volume is 924 m^3 .…
- The circumference of the circular base of a cylinder is 44 cm and its height is 15 cm.…
- The area of the base of a circular cylinder is 35 cm^2 and its height is 8 cm. The…
- A cuboid having dimensions 16 m x 11 m x 8 m is melted to form a cylinder of radius 4…
- The dimensions of a cuboid are 8m x 6 m x4 m. Its lateral surface area isA. 210 m^2 B.…
- The length, breadth and height of a cuboid are in the ratio 3 : 4 : 6 and its volume…
- The surface area of a cube is 384 cm^2 . Its volume isA. 512 cm^3 B. 256 cm^3 C. 384…
- Fill in the blanks: (i) If l, b, h be the length, breadth and height of a cuboid, then…
Exercise 20a
Question 1.Find the volume, lateral surface area and the total surface area of the cuboid whose dimensions are:
(i) length = 22 cm, breadth = 12 cm and height = 7.5 cm
(ii) length = 15 m, breadth = 6 m and height = 9 dm
(iii) length = 24 m, breadth = 25 cm and height = 6 m
(iv) length = 48 cm, breadth = 6 dm and height = 1 m
Answer:
(i) We know that,
Volume of cuboid = Length × Breadth × Height
= (22 × 12 × 7.5)
= 1980 cm3
We also know that,
Total Surface Area of cuboid = 2 (lb + bh + hl)
= 2 (22 × 12) + (22 × 7.5) + (12 × 7.5)
= 2 (264 + 165 + 90)
= 1038 cm2
Now,
Lateral surface area of cuboid = [2 (l + b) × h]
= 2 (22 + 12) × 7.5
= 510 cm2
(ii) We know that,
Volume of cuboid = Length × Breadth × Height
= (15 × 6 × 0.9)
= 81 m3
We also know that,
Total Surface Area of cuboid = 2 (lb + bh + hl)
= 2 (15 × 6) + (15 × 0.9) + (6 × 0.9)
= 2 (90 + 13.5 + 5.4)
= 217.8 m2
Now,
Lateral surface area of cuboid = [2 (l + b) × h]
= 2 (15 + 6) × 0.9
= 37.8 m2
(iii) We know that,
Volume of cuboid = Length × Breadth × Height
= (24 × 0.25 × 6)
= 36 m3
We also know that,
Total Surface Area of cuboid = 2 (lb + bh + hl)
= 2 (24 × 0.25) + (24 × 6) + (0.25 × 6)
= 2 (6 + 144 + 1.5)
= 303 m2
Now,
Lateral surface area of cuboid = [2 (l + b) × h]
= 2 (24 + 0.25) × 6
= 291 m2
(iv) We know that,
Volume of cuboid = Length × Breadth × Height
= (0.48 × 0.6 × 1)
= 0.288 m3
We also know that,
Total Surface Area of cuboid = 2 (lb + bh + hl)
= 2 (0.48 × 0.6) + (0.48 × 1) + (0.6 × 1)
= 2 (0.288 + 0.48 + 0.6)
= 2.736 m2
Now,
Lateral surface area of cuboid = [2 (l + b) × h]
= 2 (0.48 + 0.6) × 1
= 2.16 m2
Question 2.
The dimensions of a rectangular water tank are 2 m 75 cm by 1 m 80 cm by 1 m 40 cm. How many litres of water does it hold when filled to the brim?
Answer:
We know that,
1m = 100 cm
Therefore,
Dimensions of the tank will be: 2m 75cm × 1m 80 cm × 1m 40cm
= 275 cm × 180 cm × 140 cm
We know that,
Volume of cuboid = Length × Breadth × Height
= 275 × 180 × 140
= 6930000 cm3
We also know that,
1000 cm3 = 1L
Therefore,
Volume = 
= 6930 Litres
Question 3.
A solid rectangular piece of iron measures 1.05 m x 70 cm x 1.5 cm. Find the weight of this piece in kilograms if 1 cm3 of iron weighs 8 grams.
Answer:
We know that,
1m = 100cm
Therefore,
Dimensions of the iron piece will be: 105 cm × 70 cm × 1.5 cm
We know that,
Volume of cuboid = Length × Breadth × Height
Total volume of the piece of iron = 105 × 70 × 1.5
= 11025 cm3
We also know that,
1 cm3 = 8 gms
Therefore,
Weight of the piece = 11025 × 8
= 88200 g
= 
= 88.2 kg
Question 4.
The area of a courtyard is 3750 m2. Find the cost of covering it with gravel to a height of 1 cm if the gravel costs Rs. 6.40 per cubic metre.
Answer:
We know that,
1cm = 0.01m
Therefore,
Volume of the gravel used = Area × Height
= 3750 × 0.01
= 37.5 m3
It is given in the question that cost of the gravel is Rs. 6.40 per cubic meter
Therefore,
Total cost of covering = (37.5 × 6.4)
= Rs. 240
Question 5.
How many persons can be accommodated in a hall of length 16 m, breadth 12.5 m and height 4.5 m, assuming that 3.6 m3 of air is required for each person?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Total volume of hall = 16 × 12.5 × 4.5
= 900 m3
It is given in the question that 3.6 m3 of air is required for each person
Therefore,
Total number of persons that can be accommodated in the hall = 
= 
= 250 people
Question 6.
A cardboard box is 1.2 m long, 72 cm wide and 54 cm high. How many bars of soap can be put into it if each bar measures 6 cm x 4.5 cm x 4 cm?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Firstly, we have to find out volume of cardboard box
Volume of cardboard box = 120 × 72 × 54
= 466560 cm3
Now,
Volume of each bar of soap = 6 × 4.5 × 4
= 108 cm3
Therefore,
Total number of bars of soap that can be accommodated in that box = 
= 
= 4320 bars
Question 7.
The size of a matchbox is 4 cm x 2.5 cm x 1.5 cm. What is the volume of a packet containing 144 matchboxes? How many such packets can be placed in a carton of size 1.5 m x 84 cm x 60 cm?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Firstly, we have to find out volume occupied by a single matchbox
Volume occupied by a single matchbox = (4 × 2.5 × 1.5)
= 15 cm3
Now,
Volume of a packet containing 144 matchboxes = (15 × 144)
= 2160 cm3
Also,
Volume of carton = (150 × 84 × 60)
= 756000 cm3
Therefore,
Total number of packets that can be placed in a carton = 
= 
= 350 packets
Question 8.
How many planks of size 2 m x 25 cm x 8 cm can be prepared from a wooden block 5 m long, 70 cm broad and 32 cm thick, assuming that there is no wastage?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Total volume of the block = (500 × 70 × 32)
= 1120000 cm3
Total volume of each plank = 200 × 25 × 8
= 40000 cm3
Hence,
Total number of planks that can be made = 
= 
= 28 planks
Question 9.
How many bricks, each of size 25 cm x 13.5 cm x 6 cm, will be required to build a wall 8 m long, 5.4 m high and 33 cm thick?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Firstly,
Volume of the brick = 25 × 13.5 × 6
= 2025 cm3
Now,
Volume of the wall = 800 × 540 × 33
= 14256000 cm3
Hence,
Total number of bricks required = 
= 
= 7040 bricks
Question 10.
A wall 15 m long, 30 cm wide and 4 m high is made of bricks, each measuring 22 cm x 12.5 cm x 7.5 cm. If 
of the total volume of the wall consists of mortar, how many 12bricks are there in the wall?
Hint. Volume of bricks in the wall = {(1500 x 30 x 400) 
x (1500 x 30 x 400)1 cm3.
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Volume of the wall = 1500 × 30 × 400
= 18000000 cm3
Total quantity of mortar = 
= 1500000 cm3
Therefore,
Volume of bricks = 18000000 – 1500000
= 16500000 cm3
Now,
Volume of a single brick = 22 × 12.56 × 7.5
= 2062.5 cm3
Therefore,
Total number of bricks = 
= 
= 8000 bricks
Question 11.
Find the capacity of a rectangular cistern in litres whose dimensions are 11.2 m x 6 m x 5.8 m. Find the area of the iron sheet required to make the cistern.
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Volume of the cistern = 11.2 × 6 × 5.8
= 389.76 m3
= 389.76 × 1000
= 389760 litres
Now,
Area of iron sheet that is required to make the cistern = Total surface area of the cistern
We also know that,
Total Surface Area of cuboid = 2 (lb + bh + hl)
= 2 (11.2 × 6 + 11.2 × 5.8 + 6 × 5.8)
= 2 (67.2 + 64.96 + 34.8)
= 333.92 cm2
Question 12.
The volume of a block of gold is 0.5 m3. If it is hammered into a sheet to cover an area of 1 hectare, find the thickness of the sheet.
Answer:
It is given that,
Volume of the block = 0.5 m3
We know that,
1 hectare = 10000 m2
Therefore,
Thickness of the sheet = volume/area
= 0.5/10000
= 0.00005 m
= 0.005 cm
= 0.05 mm
Question 13.
The rainfall recorded on a certain day was 5 cm. Find the volume of water that fell on a 2-hectare field.
Answer:
It is given that,
Rain recorded in a certain day = 5 cm = 0.05 m
Area of the field = 2 hectare
= 2 × 10000 m2
= 20000m2
Therefore,
Total rain over the field = Area of the field × Height of the field
= 0.05 × 20000
= 1000 m3
Question 14.
A river 2 m deep and 45 m wide is flowing at the rate of 3 km/h. Find the quantity of water that runs into the sea per minute.
Answer:
It is given in the question that,
Area of cross section of river = 45m × 2m = 90 m2
Rate of flow = 3 km/hr
= 50 m/min
Therefore,
Volume of water flowing through the cross-section in one minute = 90 m2 × 50 m/min
= 4500 m3 per minute
Question 15.
A pit 5 m long and 3.5 m wide is dug to a certain depth. If the volume of earth taken out of it is 14 m3, what is the depth of the pit?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Let the depth of the pit be x m
Therefore,
Volume = 5 × 3.5 × x
It is given in the question that,
Volume = 14 m3
Therefore,
Depth, x = 
x = 
= 0.8 m
= 80 cm
Question 16.
A rectangular water tank is 90 cm wide and 40 cm deep. If it can contain 576 litres of water, what is its length?
Answer:
It is given that,
Capacity of the water tank = 576 Litres = 0.576 m3
Width = 90 cm = 0.9 m
Depth = 40 cm = 0.4 m
Therefore,
Length = 
= 1.600 m
Question 17.
A beam of wood is 5 m long and 36 cm thick. It is made of 1.35 m3 of wood. What is the width of the beam?
Answer:
It is given in the question that,
Volume of the beam = 1.35 m3
Length = 5 m
Thickness = 36 cm = 0.36 m
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Width = 
= 0.75 m
= 75 cm
Question 18.
The volume of a room is 378 m3 and the area of its floor is 84 m2. Find the height of the room.
Answer:
We know that,
Volume = Height × Area
Given that,
Volume = 378 m3
Area = 84 m2
Therefore,
Height = 
= 
= 4.5 m
Question 19.
A swimming pool is 260 m long and 140 m wide. If 54600 cubic metres of water is pumped into it, find the height of the water level in it.
Answer:
It is given in the question that,
Length of the pool = 260 m
Width of the pool = 140 m
Also,
Volume of water in the pool = 54600 cubic metres
Therefore,
Height of water = 
= 
= 1.5 metres
Question 20.
Find the volume of wood used to make a closed box of outer dimensions 60 cm x 45 cm x 32 cm, the thickness of wood being 2.5 cm all around.
Answer:
Given that,
External length = 60 cm
External width = 45 cm
External height = 32 cm
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
External volume of the box = 60 × 45 × 32
= 86400 cm3
It is also given that,
Thickness of the wood = 2.5 cm
Therefore,
Internal length = 60 – (2.5 × 2) = 55 cm
Internal width = 45 – (2.5 × 2) = 40 cm
Internal height = 32 – (2.5 × 2) = 27 cm
As we know that,
Volume of cuboid = Length × Breadth × Height
Thereore,
Internal volume of the box = 55 × 40 × 27
= 59400 cm3
Hence,
Volume of wood = External volume – Internal volume
= 86400 – 59400
= 27000 cm3
Question 21.
Find the volume of iron required to make an open box whose external dimensions are 36 cm x 25 cm x 16.5 cm, the box being 1.5 cm thick throughout. If 1 cm3 of iron weighs 8.5 grams, find the weight of the empty box in kilograms.
Answer:
Given that,
External length = 36 cm
External width = 25 cm
External height = 16.5 cm
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
External volume of the box = 36 × 25 × 16.5
= 14850 cm3
It is also given that,
Thickness of iron = 1.5 cm
Therefore,
Internal length = 36 – (1.5 × 2) = 33 cm
Internal width = 25 – (1.5 × 2) = 22 cm
Internal height = 16.5 – 1.5 = 15 cm (As the box is open)
As we know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Internal volume of the box = 33 × 22 × 15
= 10890 cm3
Hence,
Volume of iron = External volume – Internal volume
= 14850 – 10890
= 3960 cm3
Also given that,
1 cm3 of iron = 8.5 grams
Therefore,
Total weight of the box = 3960 × 8.5
= 33660 grams
= 33.66 kilograms
Question 22.
A box with a lid is made of wood which is 3 cm thick. Its external length, breadth and height are 56 cm, 39 cm and 30 cm respectively. Find the capacity of the box. Also find the volume of wood used to make the box.
Answer:
Given that,
External length = 56 cm
External width = 39 cm
External height = 30 cm
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
External volume of the box = 56 × 39 × 30
= 65520 cm3
It is also given that,
Thickness of the wood = 3 cm
Therefore,
Internal length = 56 – (3 × 2) = 50 cm
Internal width = 39 – (3 × 2) = 33 cm
Internal height = 30 – (3 × 2) = 24 cm
As we know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Capacity of box = Internal volume of the box = 55 × 40 × 27
= 39600 cm3
Hence,
Volume of wood = External volume – Internal volume
= 65520 – 39600
= 25920 cm3
Question 23.
The external dimensions of a closed wooden box are 62 cm, 30 cm and 18 cm. If the box is made of 2-cm-thick wood, find the capacity of the box.
Answer:
Given that,
External length = 62 cm
External width = 30 cm
External height = 18 cm
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
External volume of the box = 62 × 30 × 18
= 33480 cm3
It is also given that,
Thickness of the wood = 2 cm
Therefore,
Internal length = 62 – (2 × 2) = 58 cm
Internal width = 30 – (2 × 2) = 26 cm
Internal height = 18 – (2 × 2) = 14 cm
As we know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Capacity of box = Internal volume of the box = 58 × 26 × 14
= 21112 cm3
Question 24.
A closed wooden box 80 cm long, 65 cm wide and 45 cm high, is made of 2.5-cm-thick wood. Find the capacity of the box and its weight if 100 cm3 of wood weighs 8 g.
Answer:
Given that,
External length = 80 cm
External width = 65 cm
External height = 45 cm
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
External volume of the box = 80 × 65 × 45
= 234000 cm3
It is also given that,
Thickness of the wood = 2.5 cm
Therefore,
Internal length = 80 – (2.5 × 2) = 75 cm
Internal width = 65 – (2.5 × 2) = 60 cm
Internal height = 45 – (2.5 × 2) = 40 cm
As we know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Capacity of box = Internal volume of the box = 75 × 60 × 40
= 180000 cm3
Hence,
Volume of wood = External volume – Internal volume
= 234000 – 180000
= 54000 cm3
It is also given that,
100 cm3 of wood weighs 8 g
Therefore,
Weight of wood = 
= 4320 g
= 4.32 kg
Question 25.
Find the volume, lateral surface area and the total surface area of a cube each of whose edges measures:
(i) 7 m (ii) 5.6 cm (iii) 8 dm 5 cm
Answer:
(i) We have,
Length of the edge of the cube = a = 7 cm
We know that,
Volume of cube = a3 = 73 = 343 m3
Also,
Lateral surface area of the cube = 4a2
= 4 × 7 × 7
= 196 m2
Total surface area of the cube = 6a2
= 6 × 7 × 7
= 294 m2
(ii) We have,
Length of the edge of the cube = a = 5.6 cm
We know that,
Volume of cube = a3 = (5.6)3 = 175.616 cm3
Also,
Lateral surface area of the cube = 4a2
= 4 × 5.6 × 5.6
= 125.44 cm2
Total surface area of the cube = 6a2
= 6 × 5.6 × 5.6
= 188.16 cm2
(iii) We have,
Length of the edge of the cube = a = 8 dm 5 cm = 85 cm
We know that,
Volume of cube = a3 = 853 = 614125 cm3
Also,
Lateral surface area of the cube = 4a2
= 4 × 85 × 85
= 28900 cm2
Total surface area of the cube = 6a2
= 6 × 85 × 85
= 43350 cm2
Question 26.
The surface area of a cube is 1176 cm2. Find its volume.
Answer:
Let us assume the edge of the cube be a
We know that,
Total surface area of the cube = 6a2
6a2 = 1176 cm2
a = 
a = 
a = 14 cm
We also know that,
Volume of the cube = a3 = (14)3
= 2744 cm3
Question 27.
The volume of a cube is 729 cm3. Find its surface area.
Answer:
Let us assume the edge of the cube be a
We know that,
Volume of the cube = a3
a3 = 729 cm3
a = 
a = 9 cm
We also know that,
Total surface area of cube = 6a2
= 6 × 9 × 9
= 486 cm2
Question 28.
The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm. It is melted and recast into cubes, each of side 45 cm. How many cubes are formed?
Answer:
We know that,
1 m = 100 cm
Also,
Volume of a cuboid = Length × Breadth × Height
Therefore,
Volume of the original block = 225 × 150 × 27
= 911250 cm3
Given that,
Length of the edge of the cube = 45 cm
Therefore,
Volume of one cube = a3 = (45)3
= 91125 cm3
Hence,
Total number of blocks that can be cast = 
= 
= 10
Question 29.
If the length of each edge of a cube is doubled, how many times does its volume become? How many times does its surface area become?
Answer:
Let us assume a be the length of the edge of the cube
We know that,
Volume of cube = a3
Also,
Total surface area of the cube = 6a2
Now, if the length is doubled, then the new length becomes 2a
Now,
New volume = (2a)3 = 8a3
Also,
New surface area = 6 (2a)2 = 24 a2
Therefore,
The total volume of the cube is increased by the actor of 8 whereas the surface area is increased by the factor of 4.
Question 30.
A solid cubical block of fine wood costs 256 at 500 per m3. Find its volume and the length of each side.
Answer:
It is given that,
Cost of wood = Rs. 500/m3
Also,
Cost of the given block = Rs 256
We know that,
Volume of cube = a3
Therefore,
Volume of the given block = a3 = 
= 0.512 m3
= 512000 cm3
Also,
Length of its edge = a = 
= 0.8 m
= 80 cm
Exercise 20b
Question 1.Find the volume, curved surface area and total surface area of each of the cylinders whose dimensions are:
(i) radius of the base = 7 cm and height = 50 cm
(ii) radius of the base = 5.6 m and height = 1.25 m
(iii) radius of the base = 14 dm and height = 15 m
Answer:
(i) At first,
In order to find volume, we will use the following formula:
Volume of a cylinder = 
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Volume of the cylinder = 
= 
= 22 × 7 × 50
= 7700 cm3
Now,
In order to find curved surface area, we will use the following formula:
Curved surface area of cylinder = 
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Curved surface area of cylinde
r =
= 
= 22× 2× 50
= 2200 cm2
Now,
In order to find the total surface area we will use the following formula:
Total surface area of cylinder = 
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Total surface area of cylinder =
= 
= 22× 2× 57
= 2508cm2
(ii) At first,
In order to find volume we will use the following formula:
Volume of a cylinder =
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Volume of the cylinder = 
= 
= 22× 0.8× 7× 50
= 123.2 cm3
Now,
In order to find curved surface area we will use the following formula:
Curved surface area of cylinder =
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Curved surface area of cylinder =
= 
= 22× 2× 0.8× 1.25
= 44 cm2
Now,
In order to find the total surface area we will use the following formula:
Total surface area of cylinder =
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Total surface area of cylinder =
= 
= 22× 2× 0.8× 6.85
= 241.12 cm2
(iii) At first,
We will convert the radius into metre
Radius = 14dm = 1.4m
Now,
In order to find volume we will use the following formula:
Volume of a cylinder =
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Volume of the cylinder =
= 
= 22× 0.2× 1.4× 1.5
= 92.4cm3
Now,
In order to find curved surface area we will use the following formula:
Curved surface area of cylinder =
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Curved surface area of cylinder =
= 
= 22× 2× 0.2× 1.5
= 132cm2
Now,
In order to find the total surface area we will use the following formula:
Total surface area of cylinder =
Where,
‘r’ = radius of the base
‘h’ = height of the cylinder
Hence,
Total surface area of cylinder =
= 
= 22× 2× 0.2× 2.9
= 144.32cm2
Question 2.
A milk tank is in the form of a cylinder whose radius is 1.5 m and height is 10.5 m. Find the quantity of milk in litres that can be stored in the tank.
Answer:
It is given in the question that,
Radius of the cylindrical milk tank (r) = 1.5m
Height of the cylindrical milk tank (h) = 10.5m
Now,
In order to find the capacity of the tank we’ll find the volume of the milk tank
Hence,
Volume of the cylindrical milk tank = =
= 
= 
= 74.25 m3
Now,
We know that,
1 m3 = 1000 L
∴74.25 m3= 74250 L
Question 3.
A wooden cylindrical pole is 7 m high and its base radius is 10 cm. Find its weight if the wood weighs 225 kg per cubic metre.
Answer:
It is given in the question that,
Radius of the cylindrical pole (r) = 10cm = 0.1m
Height of the cylindrical pole (h) = 7m
Now,
Volume of the cylindrical wooden pole =
= 
= 
= 0.22 cm3
Now,
We know that,
Weight of the wood = 225 kg/m3
∴ Weight of the pole = 0.22× 225
= 49.5 kg
Question 4.
Find the height of the cylinder whose volume is 1.54 m3 and diameter of the base is 140 cm?
Answer:
It is given in the question that,
Volume of cylinder = 1.54 m3
Diameter of the base = 140 cm = 1.4m
Hence,
Radius of the base = 
= 0.7 m
Now,
Volume of cuboid =
1.54 = 
1.54 = 
h = 1 m
Hence, height of the cuboid = 1m
Question 5.
The volume of a circular iron rod of length 1 m is 3850 cm3. Find its diameter.
Answer:
It is given in the question that,
Volume of cylindrical rod = 3850 cm3
Height of the rod = 1m = 100cm
Now,
In order to find the diameter of the rod we need to find the radius of the rod
Hence,
Volume of cuboid =
3850 = 
3850 = 
r2 = 
r = 1.75× 7
r = 3.5 cm
Hence,
Diameter = 2(radius)
= 2 × 3.5
= 7 cm
Question 6.
A closed cylindrical tank of diameter 14 m and height 5 m is made from a sheet of metal. How much sheet of metal will be required?
Answer:
It is given in the question that,
Diameter of the cylindrical tank = 14 m
Radius of the cylindrical tank = 
= 7 m
Height of the cylindrical tank = 5 m
Now,
In order to find the total area of the metal sheet required we need to find the total surface area of the tank.
Hence,
Total surface area of the cylindrical tank = 
= 2 × 
= 
= 528 m2
Question 7.
The circumference of the base of a cylinder is 88 cm and its height is 60 cm. Find the volume of the cylinder and its curved surface area.
Answer:
It is given in the question that,
Circumference of the base of the cylinder = 88cm
Height of the cylinder = 60cm
Hence,
Curved surface area = Circumference× height
= 88 × 60
= 5280 cm2
Now,
The circumference of the base = 2
= 88 cm
Hence,
The radius of the base(r) = 
=
= 14cm
Hence,
We can find the volume as follows:
Volume of the cylinder =
= 
= 22× 2× 14× 60
= 36960cm3
Question 8.
The lateral surface area of a cylinder of length 14 m is 220 m2. Find the volume of the cylinder.
Answer:
In the question it is given that
Length of the cylinder = 14 m
Which means,
That the height of the cylinder = 14m
Lateral surface area of the cylinder = 2
220 = 
r = 
r = 2.5 m
Hence,
We can find the volume as,
Volume of the cylinder =
= 
= 22× 2× 2.5× 2.5
= 275 m3
Question 9.
The volume of a cylinder of height 8 cm is 1232 cm3. Find its curved surface area and the total surface area.
Answer:
It is given in the question that,
Height of the cylinder = 8cm
And,
Volume of the cylinder = 
Hence,
We can find the radius as,
r = 
r = 
r = √49
r = 7 cm
Now,
Curved surface area of the cylinder =
= 
= 252 cm2
Therefore,
The total surface area of the cylinder =
= 
= 2× 22× 15
= 660 cm2
Question 10.
The radius and height of a cylinder are in the ratio 7 : 2. If the volume of the cylinder is 8316 cm 3, find the total surface area of the cylinder.
Answer:
It is given in the question that,
The ratio of radius and height is 7 : 2
This means that,
r = 
Now,
We can find the volume of the cylinder as:
Volume of the cylinder =
8316 = 
8316 = 
h3 = 216
h = 6
Hence,
Radius, r =
= 
= 21 cm
Therefore,
Total surface area of the cylinder =
= 
= 2× 22× 3× 27
= 3564 cm2
Question 11.
The curved surface area of a cylinder is 4400 cm2 and the circumference of its base is 110 cm. Find the volume of the cylinder.
Answer:
In the above question it is given that
Curved surface area of the cylinder = 
= 4400 cm2
And,
The circumference of the base of the cylinder = 
Now,
The height of the cylinder = h = 
= 
= 40 cm
Also,
Radius of the cylinder = r = 
= 
= 
Hence,
We can find the volume of the cylinder as:
Volume of the cylinder = 
= 
= 38500 cm3
Question 12.
A particular brand of talcum powder is available in two packs, a plastic can with a square base of side 5 cm and of height 14 cm, or one with a circular base of radius 3.5 cm and of height 12 cm. Which of them has greater capacity and by how much?
Answer:
In the above given question,
In order to find the greater capacity pack
At first, we’ll calculate the volume of the cubic pack,
Length of the side of pack, a = 5cm
Height of the pack, h = 14cm
Hence,
Volume of the pack = a2h
= (5)2(14)
= 5× 5 × 14
= 350 cm3
Now,
We’ll calculate the volume of the cylindrical pack,
Radius of the base, r = 35cm
Height of the cylinder, h = 12 cm
Hence,
Volume of the pack =
= 
= 22× 5× 35× 12
= 462 cm3
Hence,
It’s clear that the pack with the circular has a greater capacity than the than the pack with square base.
And,
The deference between their volume= 462 – 350
= 112 cm3
Question 13.
Find the cost of painting 15 cylindrical pillars of a building at Rs. 2.50 per square metre if the diameter and height of each pillar are 48 cm and 7 metres respectively.
Answer:
It is given in the question that,
Diameter of the cylindrical pillars = 48 cm
Hence,
The radius of the cylindrical pillars =
= 24 cm
= 0.24 m
Height of the cylindrical pillars = 7 m
Now,
Lateral surface area of one pillar = 
= 
= 10.56 m2
Now,
The surface area to be painted = total surface area of 15 pillars
= 15 × 10.56
= 158.4 m2
Therefore,
The total cost of painting = Rs(158.4 × 2.5)
= Rs 396
Question 14.
A rectangular vessel 22 cm by 16 cm by 14 cm is full of water. If the total water is poured into an empty cylindrical vessel of radius 8 cm, find the height of water in the cylindrical vessel.
Answer:
It can be concluded from the question that,
Volume of the rectangular vessel = 22 × 16 × 14
= 4928 cm3
Radius of the cylindrical vessel = 4cm
Volume of the cylindrical vessel =
Now,
Since, the water is poured from the rectangular vessel to a cylindrical vessel
Therefore, the volume of the water will remain same.
Hence,
Volume of the cylindrical vessel = volume of rectangular vessel
= 4928
Height = 24.5
Question 15.
A piece of ductile metal is in the form of a cylinder of diameter 1 cm and length 11 cm. It is drawn out into a wire of diameter 1 mm. What will be the length of the wire so obtained?
Answer:
It is given in the question that,
Diameter of the wire = 1cm
Hence,
Radius of the wire = 0.5 cm
Length or the height of the wire = 11 cm
Hence,
The volume of the wire =
= 
= 8.643 cm3
Now,
We know that,
The volumes of both the cylinders would be the same.
And,
Diameter of the new wire = 1mm = 0.1 cm
Radius = 0.05cm
Therefore the new length of the wire would be = 
= 
= 1100.02 cm
= 11 m
Question 16.
A solid cube of metal each of whose sides measures 2.2 cm is melted to form a cylindrical wire of radius 1 mm. Find the length of the wire so obtained.
Answer:
It is given in the question that
Length of the edge, a = 2.2 cm
Hence,
Volume of the cube = a3
= (2.2)3
= 10.648 cm3
Now,
Volume of the wire =
Radius of the wire = 1mm = 0.1cm
We know that,
Volume of the cube = volume of the wire
Hence,
Length of the wire = 
= 
= 338.8 cm
Question 17.
How many cubic metres of earth must be dug out to sink a well which is 20 m deep and has a diameter of 7 metres? If the earth so dug out is spread over a rectangular plot 28 m by 11 m, what is the height of the platform so formed?
Answer:
It is given in the question that,
Diameter = 7m
Hence,
Radius = 3.5 m
Depth = 20 m
Volume of the earth to be dug out =
= 
= 770 m3
Volume of the earth piled upon the given plot = 28× 11 × h
Therefore,
Height = 
= 
= 2.5 m
Question 18.
A well of inner diameter 14 m is dug to a depth of 12 m. Earth taken out of it has been evenly spread all around it to a width of 7 m to form an embankment. Find the height of the embankment so formed.
Hint. Required height =
Answer:
Given that,
Inner diameter = 14 cm
Therefore,
Radius = 7 cm
Also, Depth = 12 m
Therefore,
Volume of earth dug out = Πr2h
= 
= 1848 m3
It is also given that,
Width of embankment = 7 m
Therefore,
Total radius = 7 + 7 = 14 m
Volume of embankment = Total volume – Inner volume
= Πro2h – Πr12h
= Πh (ro2 – r12)
= 
h (142 - 72)
= h (196 – 49)
= 
× 147
= 21 × 22h
= 462 × h m3
Since,
Volume of embankment = Volume of earth dug out
Therefore,
1848 = 462 h
h = 
h = 4 m
Therefore,
Height of the embankment = 4 m
Question 19.
A road roller takes 750 complete revolutions to move once over to level a road. Find the area of the road if the diameter of the road roller is 84 cm and its length is 1 m.
Answer:
It is given in the question that,
Diameter = 84 cm
Hence,
Radius = 42cm
Length = 1m = 100cm
Now,
Lateral surface area =
= 
= 26400 cm2
Hence, the area of the road will be
= lateral surface area × no. of rotations
= 26400 × 750
= 19800000 cm2
= 1980 m2
Question 20.
A cylinder is open at both ends and is made of 1.5-cm-thick metal. Its external diameter is 12 cm and height is 84 cm. What is the volume of metal used in making the cylinder? Also, find the weight of the cylinder if 1 cm ' of the metal weighs 7.5 g.
Hint. External radius = 6 cm, internal radius = 4.5 cm.
Volume of metal =
.
Answer:
It is given in the question that,
Thickness of the cylinder = 1.5 cm
External diameter of the cylinder = 12cm
Hence,
Radius = 6 cm
And,
Internal radius = 4.5cm
Height = 84cm
Hence,
We have the following measurements now,
Total volume =
= 
= 9504 cm3
Inner volume of the cylinder =
= 
= 5346 cm3
Hence,
The volume of the metal = total volume – internal volume
= 9504 – 5346
= 4158 cm3
Therefore,
Weight of the iron = 4158 × 7.5
= 31.185 kg
Question 21.
The length of a metallic tube is 1 metre, its thickness is 1 cm and its inner diameter is 12 cm. Find the weight of the tube if the density of the metal is 7.7 grams per cubic centimetre.
Hint. Weight of 1 cm3 of metal = 7.7 g.
Answer:
It is given in the question that,
Length = 1m
= 100cm
Inner diameter = 12 cm
Inner Radius = 6 cm
Hence,
Inner volume =Πr12h
Thickness = 1 cm
outer radius = 7 cm
Now,
We can calculate the following measurements:
Total volume =Πr22h
Now,
Volume of the tube = total volume – inner volume
= Πr22h- Πr12h
= Πh(r22 - r12)
= 3.14 × 100 × (72 - 62)
= 3.14 × 100 × (49 - 36)
= 314 × 13
= 4082 cm3
We have,
Density of the tube = 7.7 g/cm3
Therefore,
Weight of the tube = volume × density
= 4082 × 7.7
= 31431g
= 31.43 kg
Exercise 20c
Question 1.The maximum length of a pencil that can be kept in a rectangular box of dimensions
12 cm x 9 cm x 8 cm, is
A. 13 cm
B. 17 cm
C. 18 cm
D. 19 cm
Answer:
We know that,
Length of the diagonal of the cuboid = 
= 
= 
= 
= 17 cm
Therefore, option B is correct
Question 2.
The total surface area of a cube is 150 cm2. Its volume is
A. 216 cm3
B. 125 cm3
C. 64 cm3
D. 1000 cm3
Answer:
We know that,
Total surface area of cube = 6a2
6a2 = 150 cm2
a = 
a = 
a = 5 cm
Therefore,
Volume of the cube = a3 = 53 = 125 cm3
Hence, option B is correct
Question 3.
If the capacity of a cylindrical tank is 1848 m3 and the diameter of its base is 14 m, the depth of the tank is
A. 8 m
B. 12 m
C. 16 m
D. 18 m
Answer:
Given that,
Volume of cylindrical tank = 1848 m3
Diameter = 14 m
So, Radius = 7 m
We know that,
Volume of cylinder = 
r2h
1848 = × 7 × 7 × h
h = 
h = 12 m
Hence, option B is correct
Question 4.
The volume of a cube is 343 cm3. Its total surface area is
A. 196 cm2
B. 49 cm2
C. 294 cm2
D. 147 cm2
Answer:
Given that,
Volume of cube = 343 cm3
a3 = 343 cm3
a = 
a = 7 cm
We know that,
Total surface area of cube = 6a2
= 6 × 7 × 7
= 294 cm2
Hence, option C is correct
Question 5.
The cost of painting the whole surface area of a cube at the rate of is 10 paise per cm2Rs. 264.60. Then, the volume of the cube is
A. 6859 cm3
B. 9261 cm3
C. 8000 cm3
D. 10648 cm3
Answer:
Let the side of cube be ‘a’
Hence total surface area of cube = 6a2
Cost of painting the cube = 6a2 × 10
264.6 = 60 a2
a2 =
a2 = 4.41
a = 2.1
Hence,
Volume of the cube= a3
= (2.1)3
= 9.261 cm3
Hence, option B is correct
Question 6.
How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to build a wall 8 m long, 6 m high and 22.5 cm thick?
A. 5600
B. 6000
C. 6400
D. 7200
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Volume of each brick = 25 × 11.25 × 6
= 1687.5 cm3
Volume of wall = 800 × 600 × 22.5
= 10800000 cm3
Therefore,
Number of bricks = 
= 6400
Hence, option C is correct
Question 7.
How many cubes of 10 cm edge can be put in a cubical box of 1 m edge?
A. 10
B. 100
C. 1000
D. 10000
Answer:
Volume of smaller cube = a3 = (10)3 = 1000 cm3
Volume of box = (100)3 = 1000000 cm3
Therefore,
Total number of cubes = 
= 1000
Hence, option C is correct
Question 8.
The edges of a cuboid are in the ratio 1: 2: 3 and its surface area is 88 cm2. The volume of the cuboid is
A. 48 cm3
B. 64 cm3
C. 96 cm3
D. 120 cm3
Answer:
Let a be the length of the smallest edge
Therefore,
The edges are in proportion a: 2a: 3a
We know that,
Surface area of cuboid = 2 (lb + bh + hl)
= 2 (a × 2a + a × 3a + 2a × 3a)
= 2 (2a2 + 3a2 + 6a2)
= 22a2
= 88 cm2
a = 
= 
= 2
Also,
2a = 2 × 2 = 4
And,
3a = 3 × 2 = 6
Therefore,
Volume = a × 2a × 3a
= 2 × 4 × 6
= 48 cm3
Hence, option A is correct
Question 9.
Two cubes have their volumes in the ratio 1 : 27. The ratio of their surface areas is
A. 1 : 3
B. 1 : 9
C. 1 : 27
D. None of these
Answer:
Given that,
Volumes are in the ration 1: 27
Therefore,
= 
= 
a = 
a = 
Or b = 3a
Or 
We have to find out ratio of their surface areas:
= 
= 
= 
= 
Therefore, the surface areas are in the ratio 1: 9
Hence, option B is correct
Question 10.
The surface area of a (10 cm x 4 cm x 3 cm) brick is
A. 84 cm2
B. 124 cm2
C. 164 cm2
D. 180 cm2
Answer:
We know that,
Surface area of a cuboid = 2 (lb + bh + hl)
= 2 (10 × 4 + 10 × 3 + 4 × 3)
= 2 (40 + 30 + 12)
= 164 cm2
Hence, option C is correct
Question 11.
An iron beam is 9 m long, 40 cm wide and 20 cm high. If 1 cubic metre of iron weighs 50 kg, what is the weight of the beam?
A. 56 kg
B. 48 kg
C. 36 kg
D. 27 kg
Answer:
We know that,
Volume of a cuboid = Length × Breadth × Height
= 9 × 0.4 × 0.2
= 0.72 m3
Therefore,
Weight = 0.72 × 50
= 36 kg
Question 12.
A rectangular water reservoir contains 42000 litres of water. If the length of reservoir is 6 m and its breadth is 3.5 m, the depth of the reservoir is
A. 2 m
B. 5 m
C. 6 m
D. 8m
Answer:
We know that,
Volume of a cuboid = Length × Breadth × Height
42000 L = 42 m3 (As 1 m3 = 1000 L)
Therefore,
Height (h) = 
= 
= 2 m
Hence, option A is correct
Question 13.
The dimensions of a room are (10 mx8mx 3.3 m). How many men can be accommodated in this room if each man requires 3 m3 of space?
A. 99
B. 88
C. 77
D. 75
Answer:
We know that,
Volume of a cuboid = Length × Breadth × Height
Therefore,
Volume of the room = 10 × 8 × 3.3
= 264 m3
Space required by 1 person = 3 m3
Therefore,
Total number of people that can be accommodated = 
= 88
Hence, option B is correct
Question 14.
A rectangular water tank is 3 m long, 2 m wide and 5 m high. How many litres of water can it hold?
A. 30000
B. 15000
C. 25000
D. 35000
Answer:
For this we have to find out volume of the water tank
We know that,
Volume of a cuboid = Length × Breadth × Height
Therefore,
Volume of water tank = 3 × 2 × 5
= 30 m3
= 30000 L (As, 1 m3 = 1000 L)
Hence, option A is correct
Question 15.
The area of the cardboard needed to make a box of size 25 cm x 15 cm x 8 cm will be
A. 390 cm2
B. 1390 cm2
C. 2780 cm2
D. 1000 cm2
Answer:
We know that,
Surface area of cuboid = 2 (lb + bh + hl)
Therefore,
Area of the cardboard required to make a box = 2 (23 × 15 + 15 × 8 + 25 × 8)
= 2 (375 + 120 + 200)
= 1390 cm2
Hence, option B is correct
Question 16.
The diagonal of a cube measures 
cm. Its volume is
A. 8 cm3
B. 16 cm3
C. 27 cm3
D. 64 cm3
Answer:
Given that,
Diagonal of the cube = a
= 4 cm
i.e. a = cm
Therefore,
Volume of the cube = a3 = 43
= 64 cm3
Hence, option D is correct
Question 17.
The diagonal of a cube is 
cm long. Its total surface area is
A. 243 cm2
B. 486 cm2
C. 324 cm2
D. 648 cm2
Answer:
We know that,
Diagonal of the cube = a
= 9
i.e. a = 9
Therefore,
Total surface area of the cube = 6a2
= 6 × 9 × 9
= 486 cm2
Hence, option B is correct
Question 18.
If each side of a cube is doubled then its volume
A. is doubled
B. becomes 4 times
C. becomes 6 times
D. becomes 8 times
Answer:
Let the side of the cube be a units
Original volume = a3
Now, when each side of the cube is doubled then its volume:
New side = 2a units
New Volume = (2a)3 = 8a3 cubic units
Therefore, the volume of the cube is 8 times than its original volume
Hence, option D is correct
Question 19.
If each side of a cube is doubled, its surface area
A. is doubled
B. becomes 4 times
C. becomes 6 times
D. becomes 8 times
Answer:
Let the side of the cube be “a” unit
Original Surface area = 6a2 sq units
Now, when each side of a cube is doubled than its surface area:
New surface area = 6 (2a2) sq units
= 24a2 sq units
Therefore, the surface area of the cube is 4 times than its original area
Hence, option B is correct
Question 20.
Three cubes of iron whose edges are 6 cm, 8 cm and 10 cm respectively are melted and formed into a single cube. The edge of the new cube formed is
A. 12 cm
B. 14 cm
C. 16 cm
D. 18 cm
Answer:
We know that,
Volume of cube = a3
Total Volume of cube = 63 + 83 + 103
= 216 + 512 + 1000
= 1728 cm3
Therefore,
Edge of the new cube = 
= 12 cm
Hence, option A is correct
Question 21.
Five equal cubes, each of edge 5 cm, are placed adjacent to each other. The volume of the cuboid so formed, is
A. 125 cm3
B. 375 cm3
C. 525 cm3
D. 625 cm3
Answer:
Length of the cuboid so formed = 25 cm
Breadth of the cuboid = 5 cm
Height of the cuboid = 5 cm
We know that,
Volume of cuboid = Length × Breadth × Height
= 25 × 5 × 5
= 625 cm3
Hence, option D is correct
Question 22.
A circular well with a diameter of 2 metres, is dug to a depth of 14 metres. What is the volume of the earth dug out?
A. 32 m3
B. 36 m3
C. 40 m3
D. 44 m3
Answer:
Given that,
Diameter of the circular well = 2 m
Radius = 1 m
Height = 14 m
Therefore,
Volume of cylindrical well =
= × 1 × 1 × 14
= 44 m3
Hence, option D is correct
Question 23.
The ratio of the total surface area to the lateral surface area of 20 cm and height 60 cm, is
A. 2: 1
B. 3: 2
C. 4: 3
D. 5 : 3
Answer:
We have,
= 
= 
= 
= 
= 4: 3
Therefore, option C is correct
Question 24.
The number of coins, each of radius 0.75 cm and thickness 0.2 rightcm, to be melted to make aright circular cylinder of height 8 cm and base radius 3 cm is
A. 460
B. 500
C. 600
D. 640
Answer:
Total number of coins = 
= 
= 640
Hence, option D is correct
Question 25.
66 cm3 of silver is drawn into a wire 1 mm in diameter. The length of the wire will be
A. 78 m
B. 84 m
C. 96 m
D. 108 m
Answer:
We have to find out length of the wire:
Length = 
Diameter = 1mm (Given)
Therefore,
Radius = 0.05 cm
Length = 
= 8400 cm
= 84 m
Hence, option B is correct
Question 26.
The height of a cylinder is 14 cm and its diameter is 10 cm. The volume of the cylinder is
A. 1100 cm3
B. 3300 cm3
C. 3500 cm3
D. 7700 cm3
Answer:
We know that,
Volume of cylinder = r2h
Given that,
Diameter = 10 cm
Radius = 5 cm
Height = 14 cm
Therefore,
Volume = r2h
= × 5 × 5 × 14
= 1100 cm3
Hence, option A is correct
Question 27.
The height of a cylinder is 80 cm and the diameter of its base is 7 cm. The whole surface area of the cylinder is
A. 1837 cm2
B. 1760 cm2
C. 1942 cm2
D. 3080 cm2
Answer:
We know that,
Total surface area of the cylinder = 2 (r + h)
Given that,
Diameter = 7 cm
So, Radius = 3.5 cm
Height = 80 cm
Therefore,
Total surface area = 2 × × 3.5 (3.5 + 80)
= 22 (83.5)
= 1837 cm2
Hence, option A is correct
Question 28.
The height of a cylinder is 14 cm and its curved surface area is 264 cm2. The volume of the cylinder is
A. 308 cm3
B. 396 cm3
C. 1232 cm3
D. 1848 cm3
Answer:
We know that,
Curved surface area of the cylinder = 2
264 = 2rh
r = 
r = 3 cm
We know that,
Volume of cylinder = r2h
= × 3 × 3 × 14
= 396 cm3
Hence, option B is correct
Question 29.
The diameter of a cylinder is 14 cm and its curved surface area is 220 cm2. the volume of the cylinder is
A. 770 cm3
B. 1000 cm3
C. 1540 cm3
D. 6622 cm3
Answer:
Given that,
Diameter = 14v cm
So, Radius = 7 cm
We know that,
Curved surface area of cylinder = 2
rh
220 cm2 = 2rh
h = 
h = 5 cm
Therefore,
Volume of cylinder = r2h
= 
× 7 × 7 × 5
= 770 cm3
Hence, option A is correct
Question 30.
The ratio of the radii of two cylinders is 2 : 3 and the ratio of their heights is 5 : 3. The ratio of their volumes will be
A. 4 : 9
B. 9 : 4
C. 20 : 27
D. 27 : 20
Answer:
Given that,
= 
Also,
= 
We know that,
Volume of cylinder = r2h
Therefore,
= 
= 
Therefore, the volume of given two cylinders will be in the ration 20: 27
Hence, option C is correct
Cce Test Paper-20
Question 1.Find the volume of a cube whose total surface area is 384 cm2.
Answer:
We know that,
Total surface area of a cube = 6a2
384 = 6a2
a = 
= 8 cm
Therefore,
Volume of cone = a3 = (8)3
= 512 cm3
Question 2.
How many soap cakes each measuring 7 cm 7cmx 5 cm x 2.5 cm can be placed in a box of size 56cm x 40 cm x 25 cm?
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
Therefore,
Volume of a soap cake = 7 × 5 × 2.5
= 87.5 cm3
Also,
Volume of the box = 56 × 40 × 25
= 56000 cm3
Therefore,
Number of soap cakes = 
= 640 units
Hence,
640 cakes of soap can be placed in a box of the given size
Question 3.
The radius and height of the cylinder are in the ratio 5 : 7 and its volume is 550 cm3. Find its radius and height.
Answer:
Given that,
= 
= 
We know that,
Volume of cylinder = r2h
= × 
× × h
= 550 cm3
Therefore,
h = 
= 7 cm
Therefore,
r =
= 
= 5 cm
Question 4.
Find the number of coins, 1.5 cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder with a height of 10 cm and a diameter of 4.5 cm.
Answer:
Volume of coin = r2h = 
Volume of cylinder = r2h = 
Therefore,
Total number of coins = 
= 
= 450 coins
Thus, 450 coins must be melted to form the required cylinder
Question 5.
Find the surface area of a chalk box, whose length, breadth and height are 18 cm, 10 cm and 8 cm respectively.
Answer:
Given that,
Length = 18 cm
Breadth = 10 cm
Height = 8 cm
We know that,
Total surface area of cuboid = 2 (lb + bh + hl)
= 2 (18 × 10 + 18 × 8 + 10 × 8)
= 2 (180 + 144 + 80)
= 808 cm2
Question 6.
The curved surface area of a cylindrical pillar is 264 m2 and its volume is 924 m3. Find the diameter and height of the pillar.
Answer:
We know that,
Curved surface area of cylinder = 2rh
264 = 2
r = 
r = 
m
We also know that,
Volume of cylinder = r2h
= × × × h
= 924 m3
Now,
r =
= 
= 7 m
Therefore,
Diameter of the pillar, d = 7 × 2 = 14 m
Question 7.
The circumference of the circular base of a cylinder is 44 cm and its height is 15 cm. The volume of the cylinder is
A. 1155 cm3
B. 2310 cm3
C. 770 cm3
D. 1540 cm3
Answer:
Given that,
Height = 15 cm
Circumference = 
r = 
= 7 cm
We know that,
Volume of cylinder = r2h
= 
= 2310 cm3
Hence, option B is correct
Question 8.
The area of the base of a circular cylinder is 35 cm2 and its height is 8 cm. The volume of the cylinder is
A. 140 cm3
B. 280 cm3
C. 420 cm3
D. 210 cm3
Answer:
Given that,
Area of the base of the cylinder = 35 cm2
Height = 8 cm
Therefore,
Volume = Base area × Height
= 35 × 8
= 280 cm3
Hence, option B is correct
Question 9.
A cuboid having dimensions 16 m x 11 m x 8 m is melted to form a cylinder of radius 4 m. What is the height of the cylinder?
A. 28 m
B. 14 m
C. 21 m
D. 32 m
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
= 16 × 11 × 8
= 1408 m3
Also,
Volume of cylinder = r2h = 1408 m3
Therefore,
h = 
= 28 m
Hence, option A is correct
Question 10.
The dimensions of a cuboid are 8m x 6 m x4 m. Its lateral surface area is
A. 210 m2
B. 105 m2
C. 112 m2
D. 240 m2
Answer:
We know that,
Lateral surface area of cuboid = 2 [(l + b) × h]
= 2 [(8 + 6) × 4]
= 2 (56)
= 112 m2
Hence, option C is correct
Question 11.
The length, breadth and height of a cuboid are in the ratio 3 : 4 : 6 and its volume is 576 cm3. The whole surface area of the cuboid is
A. 216 cm2
B. 324 cm2
C. 432 cm2
D 460 cm2
Answer:
We know that,
Volume of cuboid = Length × Breadth × Height
576 = 3x × 4x × 6x
576 = 72x3
x = 
= 2
Therefore,
Total surface area of cuboid = 2 (lb + bh + hl)
= 2 (3x × 4x + 4x × 6x + 6x × 3x)
= 2 (48 + 96 + 72)
= 432 cm2
Question 12.
The surface area of a cube is 384 cm2. Its volume is
A. 512 cm3
B. 256 cm3
C. 384 cm3
D. 460 cm3
Answer:
We know that,
Surface area of cube = 6a2
384 = 6a2
a = 
a = 
a = 8 cm
Therefore,
Volume of cube = a3 = 83
= 512 cm3
Hence, option A is correct
Question 13.
Fill in the blanks:
(i) If l, b, h be the length, breadth and height of a cuboid, then its whole surface area = (………… ) sq units.
(ii) If l, b, h be the length, breadth and height of a cuboid, then its lateral surface area = (…………… ) sq units.
(iii) If each side of a cube is a, then its lateral surface area is………..sq units.
(iv) If r is the radius of the base and h be the height of a cylinder, then its volume is (………….. ) cubic units.
(v) If r is the radius of the base and h be the height of a cylinder, then its lateral surface area is (…………. ) sq units.
Answer:
(i) We know that,
Total surface area of the cuboid = 2 (lb + bh + hl
(ii) We know that,
Lateral surface area of cuboid = 2 [(l + b) × h]
(iii) We know that,
Lateral surface area of cube = 4a2
(iv) We know that,
Volume of cylinder = 2h
(v) We know that,
Lateral surface area of cylinder = 2